The Bicycle Museum of America in New Bremen, Ohio. Very interesting little museum, even if you’re not into biking. (This is about an hour north of the US Air Force Museum in Dayton. )
Also, not really hidden gems, but I highly recommend the presidential museums/libraries.
79 How about the Haus am Checkpoint Charlie? I would like to see the DDR Museum and also visit the former Stasi headquarters on Normannenstrasse.
@atomom There is also a bicycle museum in Pittsburgh you could check out. Bicycle Heaven. They say they are the “world’s largest bicycle museum & shop.”. https://www.bicycleheaven.org
Less well known than the glass flowers at Harvard, my favorite museum there is the Collection of Historical Scientific Instruments:
The pinball museum is the best way to have fun putting coins into machines in Vegas!
In the same vein, and also free, is the Musee Mechanique in San Francisco:
http://www.museemecaniquesf.com/
I second the surgical museum in Chicago; it is unknown and AMAZING. (Also, it’s walking distance from my home, so if you go, send me a PM and we can have lunch.)
I wouldn’t call the WWI museum in Kansas City a hidden gem – it’s a major tourist attraction there – but it is Smithsonian-level, breathtaking. It’s worth a special trip to Kansas City just to visit.
Another vote for Storm King. I would stuff the ballot box for Storm King. I just love going there. And if you get tired of it, or it starts to rain too hard, you can visit Dia: Beacon in Beacon NY, about 10 miles and the width of the Hudson away from Storm King.
The International Center of Photography in New York City. It’s in the process of moving, I think.
Closer to home, I don’t think anyone yet has mentioned the Brandywine Museum in Chadds Ford PA, with a core collection of Wyeths, and Winterthur in northern Delaware, essentially the du Pont family collection. The paintings are OK, but the furniture and other material culture is sensational.
In Rome, the Museo Nazionale Romano, Palazzo Massimo alle Terme, which has stunning mosaics and murals from the Villa Livia, a house owned by the wife of Emperor Augustus. I’m not going to say it’s better than the Borghese, but we enjoyed it alot.
In Barcelona, the Fundacio Tapies. Antoni Tapies was the greatest living Spanish artist pretty much from the death of Picasso until his own death five years ago, but was comparatively little known in the US. He had an impressive body of work, and his personal museum is located in a restored modernist factory building designed by one of Gaudi’s main contemporary competitors, Lluís Domènech i Montaner.
In Buffalo, NY, the Albright-Knox Art Gallery has a great collection of art from the second half of the 20th Century.
Winterthur was the site of the Downton Abbey costume collection a couple years ago – Winterthur got HUGE press (and I made two trips up there). Used to live in King of Prussia, Norristown and West Philly and never got there or to Longwood Gardens.
The Checkpoint Charlie Museem was also cool, but it’s more well-known.
Hanna, I just never expected interesting museums in KC, of all places (and I say that having lived in the area and even though most of my dad’s family lives within 50 miles of KC!).
I feel a road trip coming on…so many great ideas here!
Fort Ligonier , about an hour east of Pittsburgh, has a nice museum. Ligonier is a great town to visit. http://www.fortligonier.org
We went to Ligonier last Saturday night for dinner! (We went to Fallingwater that day.) The fort was already closed, but we loved the gazebo in the diamond and DH found the ice cream shop. 
@CountingDown , I spent summers as a kid there , in Laughlintown, next to Ligonier. Still get back to Pittsburgh to visit family so make a point to visit Ligonier. I love the Diamond and the gazebo/bandstand brings back lots of memories. We always hit the Pie Shoppe in Laughlintown- I love their raisin pie!
One more thing I recall from the Spam Museum is that Spam was invented and and still owned and produced by the Hormel Food Company. Most people, including the company’s own TV ads, pronounce Hormel as “Hor-MEL,” with the accent on the second syllable. But the Hormel family that founded the company (and may still be major shareholders) pronounces the family name as “HORM-el,” with the accent on the first syllable (rhymes with “normal”).
Not exactly a museum, but the Simon Pearce glass factory in Vermont is a treasure. The glass is beautiful and the collections are museum-worthy. You can watch the glass blowing and have lunch in the lovely restaurant.
Lots of Tiffany , beautiful glass, glass blowing demonstrations, artwork, etc. at The Chrysler Museum in Norfolk, Virginia. A real gem. http://www.chrysler.org
The SHELBURNE MUSEUM in Vermont - an eclectic collection of all kinds of stuff including a riverboat (on dry land), impressionist paintings, toys, and old carriages. They used to have a complete 1950’s house but they tore it down a few years ago.
It’s not really a hidden gem, but The House on the Rock in Spring Green Wisconsin is the most unbelievably eclectic museum I’ve ever seen. Indescribable. https://www.thehouseontherock.com/PDFs/2017_Brochure.pdf
Sadly quite out of the way, but for anyone interested in music and instruments the National Music Museum in Vermillion, South Dakota. Collection include “more than 15,000 American, European, and non-Western instruments from all cultures and historical periods”. http://nmmusd.org/
“Not exactly a museum, but the Simon Pearce glass factory in Vermont is a treasure. The glass is beautiful and the collections are museum-worthy. You can watch the glass blowing and have lunch in the lovely restaurant.”
And tour the nearby Dartmouth campus while you’re in the area.
A couple of other jewels that get overshadowed in their big cities:
Chicago’s Driehaus Museum, a restored Gilded Age mansion:
New York’s Neue Galerie with early-20th-century German and Austrian art (and a fantastic restaurant):
The Chicamacomico Life Saving Station Historic Site and Museum in Rodanthe, NC is very interesting . https://www.chicamacomico.org
It is south of the more well known (and wonderful) Wright Brothers National Memorial in Kill Devil Hills, NC.
Just remembered a couple more:
For those who like Storm King, the deCordova Museum in Lincoln MA (Boston ritzy western 'burbs) also has impressive, beautiful outdoor sculptures, and a decent indoor collection as well.
In Culver City, Los Angeles CA, the Museum of Jurassic Technology. More or less an extended joke for millennials, not all of which I got, but some of which is pretty good.
After you’ve visited Dartmouth and Simon Pearce (one of my favorite places), head over the Billings Farm & Museum in Woodstock. You’ll learn about farm life in rural Vermont at the turn of the last century. It wasn’t easy. Then you can walk in Woodstock, check out the Inn and wander around the village - a perfect day.